Finite-dimensional approximation settings for infinite-dimensional Moore-Penrose inverses

After a brief survey of basic results about finite-dimensional approximation settings (such as mathematical frameworks of various projection methods, including the least-squares method, the dual least-squares method, and the Galerkin method) for in finite-dimensional Moore-Penrose inverses, this paper proceeds to a detailed study from the following aspects: For projection methods, we investigate convergence and weak convergence of their approximation setting to develop a unified theory on projection methods for infinite-dimensional Moore-Penrose inverses; this investigation yields a fundamental convergence theorem (Theorem 2.2), from which the criterion for convergence, the criterion for weak convergence, and the generalized dual least-squares method are derived. We also derive general results on the least-squares method, by which two flaws in Groestch's results are corrected. For nonprojection methods (whose approximation setting is a more general framework), we investigate weak perfect convergence of their approximation setting and provide a necessary and suffcient condition of such convergence holding (Theorem 3.2). Several examples are proposed as counterexamples to illustrate the differences between some important concepts or as concrete algorithms to show how the present work can help to analyze their behavior.